Properties, Laws and Formulae

Real Numbers

Properties of Real Numbers

For any real numbers a, b and c;

• Closure Property w.r.t. Addition

$$ \forall \; a, b \in R \quad \Rightarrow \quad a+b \in R $$

• Closure Property w.r.t. Multiplication

$$ \forall \; a, b \in R \quad \Rightarrow \quad a . b \in R $$

• Associative Property w.r.t. Addition

$$ (a + b) + c = a + (b + c) $$

• Associative Property w.r.t. Multiplication

$$ (a . b) . c = a . (b . c) $$

• Additive Identity Property

There is a unique number zero (0) called additive identity such that:

$$ a + 0 = 0 + a = a $$

• Multiplicative Identity Property

There is a unique number one (1) called multiplicative identity such that:

$$ a . 1 = 1 . a = a $$

• Additive Inverse Property

For any real number a, there exists a unique real number −a called additive inverse of a such that:

$$ a + (−a) = 0 = (−a) + a $$

• Multiplicative Inverse Property

If a $ a \neq 0 $ is a real number, there exists a unique real number $ 1 \over a $ called multiplicative inverse of a such that:

• Commutative Property w.r.t. Addition

$$ a + b = b + a $$

• Commutative Property w.r.t. Multiplication

$$ a . b = b . a $$

• Distributive Property of Multiplication over addition

$$ a (b + c) = a . b + a . c $$

or

$$ (a + b) c = a . c + b . c $$

Percentage and Financial Arithmetic

$$ Profit \;\; percentage = {profit \over cost \;\; price} \times 100 $$

$$ Loss \;\; percentage = {loss \over cost \;\; price} \times 100 $$

$$ Discount = Marked \;\; price \; − \; Selling \;\; price $$

$$ Percentage \;\; discount = {discount \over marked \;\; price} \times 100 $$

$$ Profit \;\; / \;\; Markup = Principal \times Rate \times Time $$

$$ i.e. \;\;\; I \; = \; P \; R \; T $$

$$ Profit \; / \; Markup \; Rate = \; {Markup \times 100 \over Principal \times Time} $$

Number Sequence and Patterns

For finding nth term or general term of an arithmetic sequence,

$$ a_n = a_1 + (n − 1) d $$

For finding next terms by term to term rule in arithmetic sequence,

$$ a_n = a_{n−1} + d $$

For finding nth term or general term of a geometric sequence,

$$ a_n = a_1 \; r^{n − 1} $$

Expansion and Factorization

Basic Algebraic Formulas

$$ (a + b)^2 = a^2 + 2ab + b^2 $$

$$ (a − b)^2 = a^2 − 2ab + b^2 $$

$$ a^2 − b^2 = (a + b)(a − b) $$

Manipulation of Algebraic Expression

$$ (a + b)^3 = a^3 + 3ab (a + b) + b^3 $$

$$ (a − b)^3 = a^3 − 3ab (a − b) − b^3 $$

Laws of Exponents

In general, for any non-zero integers x and y where m and n are whole numbers.

Law of Sum of Powers

$$ x^m \times x^n = x^{m + n} $$

Law of Quotient of Powers with same base

$$ x^m \div x^n = x^{m − n} $$

Law of Power of Power

$$ (x^m)^n = x^{m n} $$

Law of Power of the Product

$$ x^m \times y^m = (x \times y)^m $$

Law of Power of the Quotient

$$ x^m \div y^m = \left( {x \over y} \right)^m $$

Law of Zero Power

$$ x^0 = 1 $$

Law of Negative Power

$$ x^{−m} = {1 \over x^m} $$

Laws of Fractional Exponents

$$ x^{1 \over n} = \sqrt[n] x $$

$$ x^{m \over n} = ( x^m )^{1 \over n} = \sqrt[n] {x^m} $$

$$ x^{m \over n} = \left( x^{1 \over n} \right) ^m = \left( \sqrt[n] x \right)^m $$